Non-linearity and intersubband population inversion in quantum wire structures

Solid-State Electronics Vol. 32, No. 12, pp. 1657-1661, 1989

0038-1101/89 $3.00+0.00 Copyright © 1989 Pergamon Press plc

Printed in Great Britain. All rights reserved

Non-linearity and Intersubband Population Inverse.on :.llQuantum Wire Structures

S. Briggs, D. Jovanovic and J. P. Leburton Beckman Institute for Advanced Science and Technology and Department of Electrical Engineering and Coordinated Science Laboratory University of Illinois at Urbana-Champaign Urbana, IL 61801

ABSTRACT

We investigate carrier transport in multi subband quasi one-dimensional (1D) structures. The Monte Carlo simulation shows that due to the lack of angular randomization during scattering processes the electron system becomes rapidly nonlinear in the presence of optic phonon scattering for fields as low as 100 V/cm at room teperature. In 1D multiband systems, when the separation between quantized levels is equal to the polar optic phonon energy, resonances similar to magneto-phonon effects occur with intersubband population inversion and the possibility of far infra-red stimulated emission.

KEYWORDS Monte Carlo, One-Dimensional Systems, Electron-Phonon Interaction, Confined Structures, III-V Compounds INTRODUCTION The physics and fabrication of quasi-one dimensional (1D) artificial structures have experienced rapid progrees in the last few years. While early confined systems were limited to the observation of quantum effects at low temperature, 1D effects are now observable at higher temperatures. Quantum wires with carrier confinement below 1000 A have recently been achieved and quantum features in the transconductance of 1D field effect devices has been reported at 77K (Ismail and co-workers, 1989). Above 77K, transport is essentially determined by phonon scattering which is a strong impediment to the observance of quantum interference effects. From a device viewpoint however, high temperature operation is desirable and the interesting aspect of 1D transport is the absence of transverse degrees of freedom which limits scattering to forward and backward events. This unique feature of 1D systems makes the carrier distribution function extremely sensitive to external perturbations and causes appreciable deviations from equilibrium relatively rapidly. These deviations show up as both non-linearities in the distribution function and large fiuetutalons due to RISOPS (Resonant Inter-Subband Optic Phonon Scattering) (Briggs and Leburton, 1989) which can result in population inversion between subbands. RISOPS effects are analogous to magnetophonon resonance (Stradling and Wood, 1968) but can occur for irregularly spaced subband energies. In this communication, we discuss these novel transport phenomena in quantum wires at room temperature. Because of the complexity of the system involving multiple subbands and a nonequilibrium electron distribution, we use a Monte Carlo simulation (Briggs and Leburton, 1988; Jacoboni and Regglani, 1983). MODEL In 1D systems, electrons are constrained to move along one direction only. Our model (Briggs and Leburton, 1988) considers GaAs-AIGaAs quantum wire systems characterized by confinement conditions for which the the electronic wave functions have simple forms which facilitate the calculation of the 1D scattering rates. In one model electrons are confined in the y-direction by a GaAs-AIGaAs quantum well (QW) of width Ly while z-confinement is achieved with a gate electrode perpendicular to the QW which forms a triangular electrostatic potential with an electric field F z. Figure l a shows this model, which is similar to the V-groove quantum wire Field Effect Transistor ('V-FET) (Sakaki, 1980) or the modulation doped GaAs-AIGaAs wire structures fabricated using ion beam assisted etching (Roukee and co-workers, 1087). The other structure we have modeled is represented in Fig. lb and uses a square well of width Ly in the y direction and a parabolic (harmonic oscillator) potential in the z direction with energy level

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separation A E . This structure is similar to the grating gate structure proposed by Warren and Antoniadis (1985). In the present simulation all electrons remain in the P valley and all subbands are parabolic. We have run simulations at 300 K with Ly in the range 250-50 A , F z in the range of 30-130 kV/cm and A E varing from 6 to 36 meV.

Energy

Energy

Y

(o)

(b)

Fig. 1. a) Schematic representation of the triangular and square well potential structure, b) Representation of the square well and harmonic oscillator potential. We consider polar-optical phonons (POP) and inelastic acoustic phonons for which the energyconservation ~-function from Fermi's golden rule reduces the phase space available for scattering to a sum over the four possible final states corresponding to forward or backward emission or absorption. The rates show a large number of peaks; each peak being proportional to the density of final states and corresponding to an emission or absorption to the bottom of a subband. These peaks make the velocity and distribution functions sensitive to the energy separation between subbands, particularly between the first and second subbands. Below the P O P emission threshold, rates are somewhat smaller t h a n in bulk, while at high energies the large number of subbands enhances the rates with respect to the bulk value. M O N T E CARLO CODE We run a steady-state single-particle Monte Carlo code, and simulate 4X10 s events to obtain good convergence of the distribution functions. Because of the large number of peaks in the rates, normal methods for computing free-flight times such as self-scattering methods are inefiicient. Instead, a direct integration method is used. For a given subband u, if r is an uniformly distributed random number on [0,1], then

E/ --lnr =

~ ),u(ku(E))At(E), E-E,

(1)

where k v is the total scattering in s u b b a n d v and At is the time increment for an electron moving in uniform energy steps of size 6E from an initial energy E; to a final energy E F. In 3D simulations it is virtually impossible to store k(E) in tabular form because of the large number of possible k values, whereas in 1D systems there are only two scalar k v values for each energy and each u (tabulated earlier by the program) and hence At can be determined quickly from Newton's second law. For 1D systems direct integration compares favorably with other methods. We model multiple scattering events between twenty quantized electronic subbands using a CRAY X-MP. This method is unique in the consideration of intersubband scattering. Since scattering to each subband must be treated seperately and involves a different rate, storing all possible scattering mechanisms for twenty subbands requires too much memory. To reduce the memory requirements, only total scattering rates are computed before execution of the Monte-Carlo software. Once the code determines t h a t an electron scatters, the individual rates are computed to determine the final state. To obtain a fast execution time, the 1D matrix elements are computed using recursion relations made possible by the simple form of the wave functions in the confining potentials. This computation was completely vectorized and found to offset the overhead introduced by the "on-line" generation of scattering rates. The typical execution time for a complete simulation was on the order of 10 minutes CPU time.

RESULTS W e have simulated transport in ID systems for various longitudlnal fieldsand confinement conditions.

Non-linearity and intersubband population inversion

1659

Because of the absence of angular randomization, the electronic system shows appreciable deviations from equilibrium at relativeLy low fields. Figure 2 shows the distribution function obtained by Monte Carlo simulation and by direct solution of the Boltzman Equation (Briggs and Leburton, 1989) at 300K in a one band model with L~ m 215 A and F x m 20 kV/cm. The simulation has been performed for three different longitudinal electric fields, 50 V/cm, 100 V / c m and 200 V/cm. For all three fields, the Monte Carlo results show non-Maxwellian profiles which agree qualitatively with the Boltzman Equation solution. This behavior is mostly due to dispersionless POP's and the singular nature of the density of states in 1D systems which cause large variations of the scattering rate around the emission threshold. The simulation also shows a strong coupling between energy points separated by ~Wl,o which is evident in the quasi-periodic nature of the distribution functions. This is especially prevalent at low field (F m 50 V/cm) and low energy where the electron energy does not change significantly during each free flight between scattering events. In particular, we attribute the decrease in f(k>0) below the phonon emission threshold to the coupling by two-phonon processes (forward absorption followed by backwards emission or backward absorption fo[lowed by forward emission) with f(k<0) which is strongly depleted in the sub-threshold region. V

l

U

l

U

l

i i i

U

I

i

I

i

1

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-1

10

n 80

l

u

l

l

l

l

n 0 0

Energy (meV)

I 20

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I 40

n

I 60

I 80

LP-3o~

Fig. 2. Distribution functions for both k > 0 and k < 0 from the Boltzman Equation and Monte Carlo simulation for confinement equal to L~ -- 215/k, F z = 20 kV/cm. For the sake of clarity, the functions have been multiplied by arbitrary constants to separate the plots. From the top down the plots are: 1) Monte Carlo, F = 200 V/cm; 2) Monte Carlo, F ---~ 100 V/cm; 3) Monte Carlo, F 50 V/cm; and 4) BTE, F = 50 V/cm. We omit the Monte Carlo behavior at E = ~'w due to numerical error caused by the large scattering rate. In highly quantized 1D systems where the subband spacing is equal to the POP energy ~'wL0, the situation appears analogous to longitudinal magneto-transport with inter-subband scattering causing magnetophonon resonance. However, unlike in magnetic fields, the spatial asymmetry in the confining potentials can result in inequally spaced energy subbands. Monte Carlo simulation shows that Resonant Inter-Subband Optic Phonon Scattering (RISOPS) similar to magnetophonon resonance takes place in 1D multi-subband structures. In Fig. 3a, we show the effect of RISOPS on band occupation for four regions of k-space: 1) n + - first subband, k > 0 ; 2) n~" - first subband, k < 0 ; 3) n + - second subband, b > 0 ; 4) n + - second subband, k < 0 . The confinement conditions at resonance are Ly = 215/L and Fz =~ g0 kV/cm. These results have been obtained for the V-FET structure, however, similar results have can be observed for a parabolic confining potential. As can be seen from the figure, at resonance (E = 36 meV) the number of forwardgoing carriers in the first subband is reduced while the number of backward-going carriers remains roughly constant. In the second band, although the total number of carriers is increased the number of backward-going carriers increases by a greater amount. This implies that the displacement of the distribution function caused by the longitudinal field is minimized at resonance and that electrons are transfered from subband I to subband II. We attribute this perturbation in the carrier concentrations to POP absorption from the bottom of band I to band II. Although other scattering mechanisms are active, their scattering rates are monotonic functions of confinement throughout the resonance regime. Only absorption to band II and emission to band I show peaks at resonance. Furthermore, the maximum scattering rate occurs for transistions to the bottom of a subband and, in addition, most electrons occupy the subband bottom, where the density of states is greatest. Therefore, the maximum transfer of electrons from band I to band II is achieved when electrons at the bottom of band I can absorb a POP and scatter to the bottom of subband TL This transfer augments the number of carriers in the second baud while depleting the first band. The increased scattering rate also tends to randomize the distribution function and minimize the effect of the drifting field.

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0.50

2! . , - , - , - , - , - , - , ' . ' , ' , ' , ' , ' , LX - - Subb~ad 1, k :>0 2, D] ---- Subband 1, k <:0 - - - Subband 2, k>0 2~ --.. Subbffind 2, ~l<0

. , . , . , . , . , - , - , ' , - , ' , ' , ' , - , "

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S e p a r a t i o n between S u b b a n d s ( m e V )

Energy Separation between Subbands (meV)

Fig. 3. a) Carrier concentration as a function of resonance. 1) n ~ , the fractional occupation in subband I with k > 0 . 2) n~-, the fractional occupation in subband I with k < 0 . 3) n2+ , the fractional occupation in s u b b a n d II with k > 0 . 4) n ~ , the fractional occupation in subband II with k ~ 0 . b) Velocity as a function of resonance for the same four regions. 1) v + . 2) v ~ . 3) v + . 4) v~'. In addition to band population, we have investigated the effect of resonance on electron velocity. Figure 3b shows the effect of confinement on velocity for the same confinement conditions in each of four regions: 1) first subband, k > 0 ; 2) first subband, k < 0 ; 3) second subband, k > 0 ; 4) second subband, k <:0. As can be seen, velocities in the first band show a increase just above resonance while velocities in the second band show a sharp decrease just below resonance. To understand this effect, consider the case illustrated in Fig. 4a, where the subbands are above resonance. There is a region at the bottom of subband I in which inter-subband scattering is forbidden. This reduces the inter-subband scattering rate for subband I and consequently enhances the velocity. As resonance is approached, this region dwindles and finally vanishes at resonance. Therefore, the velocity in subband I increases above resonance due to the creation of a "forbidden region", where intersubband scattering is suppressed. When the subbands are below resonance, as in Fig. 4b, the situation is reversed. Electrons at the bottom of subband II cannot scatter to band I and electrons in s u b b a n d I cannot scatter to the bottom of subband II. This "forbidden region" results in high velocities in s u b b a n d II below resonance. A t resonance, this region disappears and velocities in subband II drop. The total velocity, which is equal to f + X v + - f - X v summed over all subband states also shows a minimum at resonance, mainly due to the reduction f 1+ compared to f ~". We have also modeled a situation with three bands, each band separated by the phonon energy and find similar effects on both the distribution function and the carrier velocities due to RISOPS.

(a)

P, t~ ER. Kx

Ii

Kx

II LP-31~O

Fig. 4. a) Scattering mechanisms above resonance. The scattering forbidden region (F.R.) reduces the absorption rate from band I. b) The analogous situation occurs below resonance, where the scattering forbidden region reduces the emission rate from band II. The hightened carrier concentration due to RISOPS can be exploited to yield a population inversion between subbands in a quantum wire (Briggs, Jovanovic, and Leburton, 1989). We focus our investiga-

Non-linearity and intersubband population inversion

1651

tion on the ease where an upper subband is in resonance with the bottom subband and a third, intermediate subband is located slightly below the upper subband. In this ease, a population inversion can be obtained between the upper and intermediate subband. We use the triangular potential structure (similar results can be obtained with the harmonic oscillator potential) with quantum well width Ly = 150 A and F~ ----10 kV]cm. The second excited z state (y = 1, z --- 3) is in resonance with the bottom subband and the y = 1, z -- 2 state is at an intermediate energy. The narrow quantum well places the excited y states at a high energy, where they have little effect on the simulation (Fig. 5a). Figure 5b shows a strong intersubband effect due to RISOPS which causes an excess of carriers in the second subband. Because the third subband is not in resonance with the first I, the distribution functions for these subbands are very similar. The simulation shows a large inversion with 24 percent in the second subband and only 4 percent in the third subband.

Energy

1

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I (y) (z)

(1) _~ "hw

~ 1°-1

Subbond

(11

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'

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6~)

~

80

I

I

100

I

120

Energy from Well Bottom (meV)

Fig. 5. a) Energy level diagrams for RISOPS with optical transition h-w between the second and first excited z-states, b) distribution functions showing population inversion. Subband 1 is the bottom subband, 2 is the y = 1,z = 3 state, and subband 3 is the y -- 1,z ----2 state. The large peak in subband 2 is probably due to numerical uncertainties. Optical matrix elements have been calculated by a~suming a linear electron density of 2 X 106 cm -1, a photon energy of 10 meV, and an effective area of 10-1°cm 2 which corresponds to a wire spacing of 1000 A . The scattering rate for photons in this structure is 4.4 X 1012s-1 which produces an absorption coefficient, c~, of 510 cm -1. .This is relatively weak but should make far infrared (FIR) stimulated emission observable in high packed quantum wire structures. ACKNOWLEDGEMENT This work was by the U.S. National Foundation under Grant No. NSF-ECS-85-10209 and the Joint Service Electronic Program. Part of the computation has been performed by using the resources of the National Center for Supercomputing Applications (NCSA). REFERENCES Briggs, Briggs, Briggs, Briggs,

S., S., S., S.,

D. Jovanovic and J. P. Leburton (1989). AppL Phys. Lett., 5..4.,2012. and J. P. Leburton (1988). Phys. Rev. B, 2~ 8183. and J. P. Leburton (1989). Superlattices and Microstructures, ~ 145. and J. P. Leburton (lg8g). Phys. Rev. B, 2~ 8025.

Ismall, K., D. A. Antoniadis and H. I. Smith (1989). Appl. Phys. Lett., ~ 1130. Jacoboni, C., L. Reggiani (1983). Rev. Mod. Phys, ~ 845. Roukes, M. L., A. Scherer, S. J. Allen, H. G. Cralghead, R. M. Ruthen, E. D.Beebe, J. P. Harbison (1987). Phys. Rev. Lett., 5~ 3011. Sakaki, H. (1980). Jap. J. Appl. Phys., L~ L735. Stradling, R., R. Wood (1988). Journal o]Physics C, 1~ 1711. Warren, A., D. Antoniadis, H. Smith and J. Melngalis (1985). IEEE Electron Device Left., EDI,a: 294.